Generation of a library of periodic grating diffraction signals

ABSTRACT

A method of generating a library of simulated-differentiation signals (simulated signals of a periodic grating includes obtaining a measured-differentiation signal (measured signal). Hypothetical parameters are associated with a hypothetical profile. The hypothetical parameters are varied within a range to generate a set of hypothetical profiles. The range to vary the hypothetical parameters is adjusted based on the measured signal. A set of simulated signals is generated from the set of hypothetical profiles.

CROSS-REFERENCE TO RELATED APPLICATION

The present application claims the benefit of earlier filed U.S.Provisional Application Ser. No. 60/233,017, entitled GENERATION OF ALIBRARY OF PERIODIC GRATING DIFFRACTION SPECTRA, filed on Sep. 15, 2000,the entire content of which is incorporated herein by reference.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present application generally relates to generatingsimulated-diffraction signals/signals for periodic gratings. Moreparticularly, the present application relates to generating a library ofsimulated-diffraction signals indicative of electromagnetic signalsdiffracting from periodic gratings.

2. Description of the Related Art

In semiconductor manufacturing, periodic gratings are typically used forquality assurance. For example, one typical use of periodic gratingsincludes fabricating a periodic grating in proximity to the operatingstructure of a semiconductor chip. The periodic grating is thenilluminated with an electromagnetic radiation. The electromagneticradiation that deflects off of the periodic grating are collected as adiffraction signal. The diffraction signal is then analyzed to determinewhether the periodic grating, and by extension whether the operatingstructure of the semiconductor chip, has been fabricated according tospecifications.

In one conventional system, the diffraction signal collected fromilluminating the periodic grating (the measured-diffraction signal) iscompared to a library of simulated-diffraction signals. Eachsimulated-diffraction signal in the library is associated with atheoretical profile. When a match is made between themeasured-diffraction signal and one of the simulated-diffraction signalsin the library, the theoretical profile associated with thesimulated-diffraction signal is presumed to represent the actual profileof the periodic grating.

The accuracy of this conventional system depends, in part, on the rangeand/or resolution of the library. More particularly, the range of thelibrary relates to the range of different simulated-diffraction signalsin the library. As such, if the collected-diffraction signal is outsideof the range of the library, then a match cannot be made. The resolutionof the library relates to the amount of variance between the differentsimulated-diffraction signals in the library. As such, a lowerresolution produces a coarser match.

Therefore, the accuracy of this convention system can be increased byincreasing the range and/or resolution of the library. However,increasing the range and/or the resolution of the library also increasesthe amount of computations required to generate the library. As such, itis desirable to determine an appropriate range and/or resolution for thelibrary without unduly increasing the amount of computations required.

SUMMARY

The present application relates to generating a library ofsimulated-diffraction signals (simulated signals) of a periodic grating.In one embodiment, a measured-diffraction signal of the periodic gratingis obtained (measured signal). Hypothetical parameters are associatedwith a hypothetical profile. The hypothetical parameters are variedwithin a range to generate a set of hypothetical profiles. The range tovary the hypothetical parameters is adjusted based on the measuredsignal. A set of simulated signals is generated from the set ofhypothetical profiles.

DESCRIPTION OF THE DRAWING FIGURES

The present invention can be best understood by reference to thefollowing description taken in conjunction with the accompanying drawingfigures, in which like parts may be referred to by like numerals:

FIG. 1 is a perspective view of a system for illuminating a periodicgrating with an incident signal and detecting deflection signals fromthe periodic grating.

FIG. 2 is a cross-sectional view of a portion of a periodic gratinghaving multiple layers;

FIG. 3 is a cross-sectional view of the multiple layers of the periodicgrating in FIG. 2 being formed separately on the substrate of theperiodic grating in FIG. 2;

FIG. 4 is a cross-sectional view of the multiple layers of the periodicgrating in FIG. 2 being formed sequentially on the substrate of theperiodic grating in FIG. 2;

FIG. 5 is a graphical depiction of an exemplary hypothetical profile ofa periodic grating;

FIG. 6 is a graph depicting the mapping of a rectangularization problemas a set-cover problem;

FIG. 7 is another graph depicting the mapping of anotherrectangularization problem as a set-cover problem;

FIGS. 8A through 8E are cross-sections of various exemplary hypotheticalprofiles of periodic gratings;

FIG. 9 is a graph of two parameters;

FIG. 10 is a signal space; and

FIG. 11 is a flow chart of an exemplary library generation process.

DETAILED DESCRIPTION

The following description sets forth numerous specific configurations,parameters, and the like. It should be recognized, however, that suchdescription is not intended as a limitation on the scope of the presentinvention, but is instead provided to provide a better description ofexemplary embodiments.

With reference to FIG. 1, a periodic grating 145 is depicted on asemiconductor wafer 140. As depicted in FIG. 1, wafer 140 is disposed ona process plate 180, which can include a chill plate, a hot plate, adeveloper module, and the like. Alternatively, wafer 140 can also bedisposed on a wafer track, in the end chamber of an etcher, in anend-station or metrology station in a chemical mechanical polishingtool, and the like.

As described earlier, periodic grating 145 call be formed proximate toor within an operating structure formed on wafer 140. For example,periodic grating 145 can be formed adjacent a transistor formed on wafer140. Alternatively, periodic grating 145 can be formed in an area of thetransistor that does not interfere with the operation of the transistor.As will be described in greater detail below, the profile of periodicgrating 145 is obtained to determine whether periodic grating 145, andby extension the operating structure adjacent periodic grating 145, hasbeen fabricated according to specifications.

More particularly, as depicted in FIG. 1, periodic grating 145 isilluminated by an incident signal 110 from an electromagnetic source120, such as an ellipsometer, reflectometer, and the like. Incidentsignal 110 is directed onto periodic grating 145 at an angle of incidentθ_(i) with respect to normal {overscore (n)} of periodic grating 145.Diffraction signal 115 leaves at an angle of θ_(d) with respect tonormal {overscore (n)}. In one exemplary embodiment, the angle ofincidence θ_(i) is near the Brewster's angle. However, the angle ofincident θ_(i) can vary depending oil the application. For example, inan alternative embodiment, the angle of incident θ_(i) is between about0 and about 40 degrees. In another embodiment, the angle of incidentθ_(i) is between about 30 and about 90 degrees. In still anotherembodiment, the angle of incident θ_(i) is between about 40 and about 75degrees. In yet another embodiment, the angle of incident θ_(i) isbetween about 50 and about 70 degrees.

Diffraction signal 115 is received by detector 170 and analyzed bysignal-processing system 190. When electromagnetic source 120 is anellipsometer, the magnitude ψ and the phase Δ of diffraction signal 115is received and detected. When electromagnetic source 120 is areflectometer, the relative intensity of diffraction signal 115 isreceived and detected.

Signal-processing system 190 compares the diffraction signal received bydetector 170 to simulated-diffraction signals stored in a library 185.Each simulated-diffraction signal in library 185 is associated With atheoretical profile. When a match is made between the diffraction signalreceived from detector 170 and one of the simulated-diffraction signalsin library 185, the theoretical profile associated with the matchingsimulated-diffraction signal is presumed to represent the actual profileof periodic grating 145. The matching simulated-diffraction signaland/or theoretical profile can then be provided to assist in determiningwhether the periodic grating has been fabricated according tospecifications.

As described above, library 185 includes simulated-diffraction signalsthat are associated with theoretical profiles of periodic grating 145.As depicted in FIG. 11, in the present exemplary embodiment, the processfor generating library 185 can include: (1) characterizing the filmstack of the periodic grating; (2) obtaining the optical properties ofthe materials used in forming the periodic grating; (3) obtainingmeasured-diffraction signals from the periodic grating; (4) determiningthe number of hypothetical parameters to use in modeling the profile ofthe periodic grating; (5) adjusting the range to vary the hypotheticalparameters in generating a set of hypothetical profiles; (6) determiningthe number of layers to use in dividing up a hypothetical profile togenerate a stimulated-diffraction signal for the hypothetical profile;(7) determining the number of harmonic orders to use in generating theset of simulated-diffraction signals; (8) determining a resolution touse in generating the set of simulated-diffraction signals; (9)generating the set of simulated-diffraction signals based on theadjusted range, parameterization, and/or resolution; and (10) comparinga set of measured-diffraction signals with the simulated-diffractionsignals in the library.

With reference to FIG. 1, the process outlined above and described ingreater detail below for generating library 185 can be performed bysignal-processing system 190. Additionally, although signal-processingsystem 190 and detector 170 and electromagnetic source 120 are depictedbeing connected by lines 126 and 125, data can be communicated betweensignal-processing system 190 and detector 170 and electromagnetic source120 through various methods and media. For example, data can becommunicated using a diskette, a compact disk, a phone line, a computernetwork, the Internet, and the like.

Furthermore, it should be noted that the process outlined above forgenerating library 185 is meant to be exemplary and not exhaustive orexclusive. As such, the process for generating library 185 can includeadditional steps not set forth above. The process for generating library185 can also include fewer steps than set forth above. Additionally, theprocess for generating library 185 can include the steps set forth abovein a different order. With this in mind, the exemplary process outlinedabove is described in greater detail below:

1. Characterizing the Film Stack of the Periodic Grating:

With continued reference to FIG. 1, prior to generating library 185,characteristics of periodic grating 145 are obtained. For example, thefollowing information can be acquired:

-   -   Specific of the measurement tool to be used, such as the        incident angle and wavelength rate of the illuminating incident        signal 110.    -   The materials used in forming periodic grating 145 and which of        the layers in the stack are patterned.    -   A range for each of the parameters for periodic grating 145,        such as the thickness in the case of un-patterned layers, or the        width (i.e., “critical dimension” or “CD”) and thickness in the        case of patterned layers.    -   A desired resolution for the critical dimension in the case of        patterned films.    -   A pitch, i.e., periodicity length, of patterned-film periodic        grating 145.    -   A specification of the type of expected profile shapes, such as        “footings”, “undercuts”, and the like.

These characteristics of periodic grating 145 can be obtained based onexperience and familiarity with the process. For example, thesecharacteristics can be obtained from a process engineer who is familiarwith the process involved in fabricating wafer 140 and periodic grating145. Alternatively, these characteristics can be obtained by examiningsample periodic gratings 145 using Atomic Force Microscope (AFM),tilt-angle Scanning Electron Microscope (SEM), X-SEM, and the like.

2. Obtaining the Optical Properties of the Materials Used in Forming thePeriodic Grating.

In the present exemplary embodiment, the optical properties of thematerials used in forming the periodic grating are obtained by measuringdiffraction signals. With reference to FIG. 2, assume for example that asample periodic grating includes four layers (i.e., layers 204, 206,208, and 210) of different materials formed on a substrate 202. Assumefor example that layers 204, 206, 208, and 210 are Gate Oxide,Polysilicon, Anti-reflective coating, and Photoresist, respectively, andthat substrate 202 is Silicon.

As depicted in FIG. 3, the optical properties of each material can beobtained by measuring a separate diffraction signal for each layer 204,206, 208, and 210 formed on substrate 202. More particularly, adiffraction signal is measured for layer 204 formed on substrate 202. Aseparate diffraction signal can be measured for layer 206 formed onsubstrate 202. Another separate diffraction signal can be measured forlayer 208 formed on substrate 202. And yet another separate diffractionsignal can be measured for layer 210 formed on substrate 202.

Alternative, as depicted in FIG. 4, in accordance with what is hereinreferred to as the “Additive Stack” approach, diffraction signals aremeasured as layers 204, 206, 208, and 210 are sequentially formed on topof substrate 202. More particularly, a diffraction signal is measuredafter forming layer 204 on substrate 202. Another diffraction signal ismeasured after forming layer 206 on layer 204. Still another diffractionsignal is measured after forming layer 208 on layer 206. Yet anotherdiffraction signal is measured after forming layer 210 on layer 208.

With reference again to FIG. 1, after measuring the diffraction signalsfor each material used to form periodic grating 145, the opticalproperties for each material is extracted. More particularly, withreference again to FIG. 2, assuming for example that periodic grating145 (FIG. 1) includes layers 204 through 210 formed on substrate 202,the optical properties of each layer 204 through 210 are extracted. Inthe present exemplary embodiment, the real and imaginary parts (n and k)of the refractive index of each layer 204 through 210 are extractedusing an optimizing engine in conjunction with a thin filmelectromagnetic equation solver. For example, the refractive index canbe extracted using a simulated annealing based optimizer, hereinreferred to as a Simulated Annealing for Continuous (SAC) variablesoptimizer.

When layers 204 through 210 include a metal layer, which is highlyreflective, the incident signal 110 (FIG. 1) may only penetrate themetal layer to a “skin depth” of typically a few nanometers. Hence, onlythe n-k can be extracted, while the nominal thickness value are notmeasured but obtained theoretically or based on experience, such as froma process engineer.

For non-metal layers, a variety of physical models can be used inconjunction with the SAC optimizer to extract the optical properties,including the thickness, of the films. For examples of suitable physicalmodels, see G. E. Jellison, F. A. Modine, “Parameterization of theoptical functions of amorphous materials in the interband region”.Applied Physics Letters, 15 vol. 69, no. 3, 371-373, July 1996, and A.R. Forouhi. I. Bloomer, “Optical Properties of crystallinesemiconductors and dielectrics”. Physical Review B., vol. 38, no. 3,1865-1874, July 1988, the entire content of which is incorporated hereinby reference.

Additionally, when an ellipsometer is used to obtain the diffractionsignals, the logarithm of the tan (ψ) signal and the cos (Δ) signal canbe compared (as is described in “Novel DUV Photoresist Modeling byOptical Thin-Film Decompositions from SpectralEllipsometry/Reflectometry Data,” SPIE LASE 1998, by Xinhui Niu, NickhilHarshvardhan Jakatdar and Costas Spanos the entire content of which isincorporated herein by reference). Comparing the logarithm of tan (ψ)and cos (Δ) rather than simply tan (ψ) and cos (Δ) has the advantage ofbeing less sensitive to noise.

3. Obtaining Measured-Diffraction Signals from the Periodic Grating

In the present exemplary embodiment, prior to generating library 185, ameasured-diffraction signal is obtained from at least one sampleperiodic grating 145. However, multiple measured-diffraction signals arepreferably obtained from multiple sites on wafer 140. Additionally,multiple measured-diffraction signals can be obtained from multiplesites on multiple wafers 140. As will be described below, thesemeasured-diffraction signals can be used in generating library 185.

4. Determining the Number of Hypothetical Parameters to Use in Modelingthe Profile of the Periodic Grating

In the present exemplary embodiment, a set of hypothetical parameters isused to model the profile of periodic grating 145 (FIG. 1). Moreparticularly, a set of hypothetical parameters is used to define ahypothetical profile, which can be used to characterize the actualprofile of periodic grating 145 (FIG. 1). By varying the values of thehypothetical parameters, a set of hypothetical profiles can begenerated.

For example, with reference to FIG. 8A, two hypothetical parameters(i.e., h1 and w1) can be used to model a rectangular profile. Asdepicted in FIG. 8A, h1 defines the height of the hypothetical profile,and w1 defines the width of the hypothetical profile. By varying thevalues of h1 and w1, a set of rectangular hypothetical profiles can begenerated.

With reference now to FIG. 8B, three hypothetical parameters (i.e., h1,w1, and t1) can be used to model a trapezoidal profile. As depicted inFIG. 8B, t1 defines the angle between the bottom and side of thehypothetical profile. Again, by varying these hypothetical parameters, aset of hypothetical profiles can be generated.

With reference now to FIG. 8C, five hypothetical parameters (i.e., w1,w2, h, p1, and w3) can be used to model a trapezoidal profile with toprounding. As depicted in FIG. 8C, w1 defines the bottom width, w2defines the top width of the trapezoidal profile, and w3 defines thewidth of the rounded top. Additionally, h defines the total height, p1defines the height of the rounded top, and the ratio p1/h defines thepercent of the height that is rounded. Again, by varying thesehypothetical parameters, a set of hypothetical profiles can begenerated.

With reference not to FIG. 8D, seven hypothetical parameters (i.e., w1,w2, p1, h, p2, w3, and w4) can be used to model a trapezoid profile withtop rounding and bottom footing. As depicted in FIG. 8D, w1 defines thewidth of the bottom footling, w2 defines the bottom width of thetrapezoidal profile, w3 defines the top width of the trapezoidalprofile, and w4 defines the width of the rounded top. Additionally, hdefines the total height, p1 defines the height of the bottom footing,and p2 defines the height of the rounded top. Thus, the ratio of p1/hdefines the percent of the height that is the bottom footing, and theratio of p2/h defines the percent of the height that is rounded. Again,by varying these hypothetical parameters, a set of hypothetical profilescan be generated.

With reference now to FIG. 8E, eight hypothetical parameters (i.e., w1,w2, p1, h1, h2, w3, w4, and d1) can be used to model a trapezoidalprofile with top rounding, bottom footing, and lateral offsets betweentwo films. As depicted in FIG. 8E, w1 defines the width of the bottomfooting, w2 defines the bottom width of the trapezoidal profile, w3defines the top width of the trapezoidal profile, and w4 defines thewidth of the top film. Additionally, h1 defines the height of the firstfilm, h2 defines the height of the 2nd film, p1 defines the height ofthe bottom footing, the ratio of p1/h1 defines the percent of the heightof the first film that is the bottom footing, and d1 defines the offsetbetween the first and second films. Again, by varying these hypotheticalparameters, a set of hypothetical profiles can be generated.

In this manner, any number of hypothetical parameters can be used togenerate hypothetical profiles having various shapes and features, suchas undercutting, footing, t-topping, rounding, concave sidewalls, convexsidewalls, and the like. It should be understood that any profile shapecould be approximated using combinations of stacked trapezoids. Itshould also be noted that although the present discussion focuses onperiodic gratings of ridges, the distinction between ridges and troughsis somewhat artificial, and that the present application may be appliedto any periodic profile.

As will be described in greater detail below, in the present embodiment,a simulated-diffraction signal can be generated for a hypotheticalprofile. The stimulated-diffraction signal can then be compared with ameasured diffraction signal from periodic grating 145 (FIG. 1). If thetwo signals match, then the hypothetical profile is assumed tocharacterize the actual profile of periodic grating 145 (FIG. 1).

The accuracy of this match depends, in part, on the selection of theappropriate number of parameters to account for the complexity of theactual profile of periodic grating 145 (FIG. 1). More particularly,using too few parameters can result in coarse matches, and using toomany parameters can unnecessarily consume time and computationalcapacity.

For example, assume that the actual profile of periodic grating 145(FIG. 1) is substantially rectangular in shape. In this case, using twoparameters, as depicted in FIG. 8A and described above, is sufficient togenerate a set of hypothetical profiles to match the actual profile ofperiodic grating 145 (FIG. 1). However, the set of hypothetical profilesgenerated using three or more parameters can include hypotheticalprofiles generated using two parameters. More particularly, when t1 is90 degrees, the hypothetical profiles generated using three parameterscan include the set of rectangular hypothetical profiles generatingusing two parameters. However, since the actual profile of periodicgrating 145 (FIG. 1) is rectangular, all of the hypothetical profilesgenerated using three parameters that are not rectangular (i.e., wheret1 is not 90 degrees) are unnecessary. However, if the actual profile ofperiodic grating 145 (FIG. 1) is trapezoidal, then using two parameterswould have resulted in a coarse match or no match.

As such, in the present exemplary embodiment, the measured-diffractionsignals that were obtained prior to generating library 185 (FIG. 1) areused to determine the appropriate number of parameters to use ingenerating library 185 (FIG. 1). More particularly, in oneconfiguration, the number of hypothetical parameters can be increaseduntil the stimulated-diffraction signal generated from the hypotheticalprofile defined by the hypothetical parameters matches themeasured-diffraction signal within a desired tolerance. One advantage ofincreasing rather than decreasing the number of hypothetical parametersused is that it can be more time and computationally efficient since thelarger sets of hypothetical profiles generated by the higher numbers ofhypothetical parameters are not always needed.

Alternatively, in another configuration, the number of hypotheticalparameters can be decreased until the simulated-diffraction signalgenerated from the hypothetical profile defined by the hypotheticalparameters ceases to match the measured-diffraction signal within adesired tolerance. One advantage of decreasing rather than increasingthe number of hypothetical parameters is that it can be more easilyautomated since the hypothetical profiles generated by lower numbers ofhypothetical parameters are typically subsets of the hypotheticalprofiles generated by higher numbers of hypothetical parameters.

Additionally, in the present exemplary embodiment, a sensitivityanalysis can be performed on the hypothetical parameters. By way ofexample, assume that a set of hypothetical parameters are used thatinclude 3 width parameters (i.e., w1, w2, and w3). Assume that thesecond width, w2, is an insensitive width parameter. As such, when thesecond width, w2, is varied, the simulated-diffraction signals that aregenerated do not vary significantly. As such, using a set ofhypothetical parameters with an insensitive parameter can result in acoarse or incorrect match between the hypothetical profile and theactual profile.

As such, in one configuration, after a match is determined between asimulated-diffraction signal and the measured-diffraction signal thatwas obtained prior to generating library 185 (FIG. 1), each hypotheticalparameter in the set of hypothetical parameters used to generate thesimulated-diffraction signal is perturbed and a newsimulated-diffraction signal is generated. The greater the effect on thenewly generated simulated-diffraction signal, the more sensitive theparameter.

Alternatively, in another configuration, after a match is determinedbetween a simulated-diffraction signal and the measured-diffractionsignal obtained prior to generating library 185 (FIG. 1), the number ofhypothetical parameters used to generate the simulated-diffractionsignal is increased or decreased by one. Assume that the number ofhypothetical parameters was being increased to determine an appropriatenumber of hypothetical parameters to use in modeling periodic grating145 (FIG. 1). In this ease, the number of hypothetical parameters isincreased by one and additional simulated-diffraction signals aregenerated. If a similar match is found between the measured-diffractionsignal and one of these simulated-diffraction signals, then theadditional hypothetical parameter is insensitive.

Assume now that the number of hypothetical parameters was beingdecreased to determine the appropriate number of parameters to use inmodeling periodic grating 145 (FIG. 1). In this case, the number ofhypothetical parameters is decreased by one and additionalsimulated-diffraction signals are generated. If a match is found betweenthe measured-diffraction and one of these simulated-diffraction signals,then the hypothetical parameters that was removed is insensitive. Thenew adjusted parameterization will exclude all parameters deemed to beinsensitive and include all parameters that were found to be sensitive.

Once the parameterization is completed, the critical dimension (CD) canthen be defined based on any portion of the profile. Following are twoexamples of CD definitions based on the profile of FIG. 8E:

-   -   Definition 1: CD=w1    -   Definition 2: CD=w1/2+(4 w2+w3)/10        CD definitions can be user specific, and the above are typical        examples that can be easily modified to suit different needs.        Thus, it should be understood that there are a wide variety of        CD definitions that will prove useful in various circumstances.

5. Adjusting the Range to Vary the Hypothetical Parameters in Generatinga Set of Hypothetical Profiles

As described above, a set of hypothetical profiles can be generated byvarying the hypothetical parameters. As will be described in greaterdetail below, a simulated-diffraction signal can be generated for eachof the hypothetical profile in this set. Thus, the range ofsimulated-diffraction signals available in library 185 (FIG. 1) isdetermined, in part, by the range within which the hypotheticalparameters are varied.

As also describe above, an initial range over which the hypotheticalparameters are to be varied can be obtained from users/customers. Insome cases, however, this initial range is based on mere conjecture.Even when this initial range is based on empirical measurements, such asmeasurements of samples using AFM, X-SEM, and the like, inaccuracy inthe measurement can produce poor results.

As such, in the present exemplary embodiment, the range over which thehypothetical parameters are to be varied is adjusted based on themeasured-diffraction signal obtained prior to generating library 185(FIG. 1). In brief, to determine the appropriateness of the range,multiple simulated-diffraction signals are generated until one matchesone of the measured-diffraction signals. When a match is found, thehypothetical parameter values that were used to generate the matchingsimulated-diffraction signal are examined. More particularly, bydetermining where in the range these hypothetical parameter values fall,the appropriateness of the range can be determined, and the range can beadjusted as needed. For example, if these hypothetical parameters falltoward one end of the range, the range can be shifted and re-centered.

In the present exemplary embodiment, the range over which thehypothetical parameters are to be varied is adjusted before generatinglibrary 185 (FIG. 1). As will be described in greater detail below, thesimulated-diffraction signals in library 185 (FIG. 1) are generatedusing the adjusted range of values for the hypothetical parameters.However, it should be understood that the range can be adjusted aftergenerating library 185 (FIG. 1), then library 185 (FIG. 1) can bere-generated using the adjusted range.

Additionally, in the present exemplary embodiment, an optimizationroutine is used to generate matching simulated-diffraction signals. Moreparticularly, a range of hypothetical parameters to be used in theoptimization process is selected. Again, if the profile shape is knownin advance, due to AFM or X-SEM measurements, a tighter range can beused. However, when the profile shape is not known in advance, a broaderrange can be used, which can increase the optimization time.

An error metric is selected to guide the optimization routine. In thepresent exemplary embodiment, the selected error metric is thesum-squared-error between the measured and simulated diffractionsignals. While this metric can work well for applications where theerror is identically and independently normally distributed (iind) anddifferences are relevant, it may not be a good metric for cases wherethe error is a function of the output value (and is hence not iind) andratios are relevant. A sum-squared-difference-log-error can be a moreappropriate error metric when the error is an exponential function ofthe output. Therefore, in the present embodiment, the sum-squared-erroris used in comparisons of cos(Δ), and thesum-squared-difference-log-error is used in comparisons of tan(ψ) wherethe ratio of the 0^(th) order TM reflectance to the zeroth order TEreflectance is given by tan(ψ)e^(iΔ).

After selecting an error metric, the optimization routine is run to findthe values of the hypothetical parameters that produce asimulated-diffraction signal that minimizes the error metric betweenitself and the measured-diffraction signal. More particularly, in thepresent exemplary embodiment, a simulated annealing optimizationprocedure is used (see “Numerical Recipes,” section 10.9, Press,Flannery, Teukolsky & Vetterling, Cambridge University Press, 1986, theentire content of which is incorporated herein by reference).Additionally, in the present exemplary embodiment, simulated-diffractionsignals are produced by rigorous models (sec University of California atBerkeley Doctoral Thesis of Xinhui Niu, “An Integrated System of OpticalMetrology for Deep Sub-Micron Lithography,” April 20, 1999, the entirecontent of which is incorporated herein by reference).

In the present exemplary embodiment, if the simulated-diffraction signalmatches the measured-diffraction signal to within a standard chi-squaredgoodness-of-fit definition (see Applied Statistics by J. Neter, W.Wasserman, G, Whitmore, Publishers: Allyn and Bacon, 2^(nd) Ed. 1982,the entire content of which is incorporated herein by reference), thenthe optimization is considered successful. The values of all of thehypothetical parameters are then examined and the CD is calculated.

This process is repeated to find matching simulated-diffraction signalsfor all of the measured-diffraction signals. The appropriateness of therange of the hypothetical parameters can then be determined by examiningwhere the values of the hypothetical parameters of the matchingsimulated-diffraction signals lie in the range. For example, if theygroup near one end of the range, then the range can be shifted andre-centered. If they lie at the limits of the range, then the range canbe broadened.

If the optimization process is unable to find a matchingsimulated-diffraction signal for a measured-diffraction signal, theneither the range or the number of hypothetical parameters need to bealtered. More particularly, the values of the hypothetical parametersare examined, and if they lie close to the limit of a range, then thisis an indication that that range should be altered. For example, therange can be doubled or altered by any desirable or appropriate amount.If the values of the hypothetical parameters do not lie close to thelimits of a range, then this typically is an indication that the numberand/or type of hypothetical parameters being used to characterize theprofile shape needs to be altered. In either case, after the range ofthe number of hypothetical parameters is altered, the optimizationprocess is carried out again.

6. Determining the Number of Layers to Use in Dividing Up a HypotheticalProfile to Generate a Simulated-Diffraction Signal for the HypotheticalProfile.

As described above, a set of hypothetical parameters defines ahypothetical profile. A simulated-diffraction signal is then generatedfor each hypothetical profile. More particularly, in the presentexemplary embodiment, the process of generating simulated-diffractionsignals for a hypothetical profile includes partitioning thehypothetical profile into a set of stacked rectangles that closelyapproximates the shape of the hypothetical profile. From the set ofstacked rectangles for a given hypothetical profile, the correspondingsimulated-diffraction signals are generated (see University ofCalifornia at Berkeley Doctoral Thesis of Xinhui Niu, “An IntegratedSystem of Optical Metrology for Deep Sub-Micron Lithography,” Apr. 20,1999, the entire content of which is incorporated herein by reference;and U.S. patent application Ser. No. 09/764,780, entitled CACHING OFINTRA-LAYER CALCULATIONS FOR RAPID RIGOROUS COUPLED-WAVE ANALYSIS, filedon Jan. 17, 2001, the entire content of which is incorporated herein byreference).

Therefore, the quality of the library depends, in part, on how well theselected sets of stacked rectangles approximate the hypotheticalprofiles. Furthermore, since a typical library 185 (FIG. 1) can includehundreds of thousands of theoretical profiles, it is advantageous torapidly automate the process of selecting a set of stacked rectanglesfor a hypothetical profile.

It should be noted that deciding on a fixed number of rectangles for aprofile without consideration of the profile shape, and thenrepresenting the profile using the fixed number of rectangles of equalheight, is not a rapid or efficient method. This is because the optimalnumber of rectangles that approximates one profile call be differentfrom the optimal number of rectangles that approximate another profile.Also, the heights of the stacked rectangles that approximate a givenprofile need not be the same. Thus, in order to provide a goodapproximation, the number of rectangles, k, and the height of therectangles are preferably determined for each profile.

However, the library generation time is a linear function of the numberof rectangles, k. Consequently, increasing k in order to improve thelibrary quality results in an increase in the amount of time required togenerate a library 185 (FIG. 1) Therefore, it is advantageous to closelyapproximate each profile with a minimum number of rectangles by allowingrectangles to have variable heights.

Thus, in one exemplary embodiment, a process is provided to determinethe number k of rectangles of varying heights that better approximatethe shape of a profile. More particularly, this problem is transformedinto a combinatorial optimization problem called a “set-cover” problem.Heuristics can then be used to solve the “set-cover” problem.

In brief, a set-cover problem involves a base set B of elements, and acollection C of sets C1, C2, . . . . Cn, where each Ci is a propersubset of B, and the sets C1, C2, . . . . Cn may share elements.Additionally, each set Ci has weight Wi associated with it. The task ofa set-cover problem is to cover all the elements in B with sets Ci suchthat their total cost, Σ_(i)Wi, is minimized.

Returning to the present application of transforming the problem ofrectangularization into a “set-cover” problem, let P denote a givenprofile. For ease of presentation, the profile P will be considered tobe symmetric along the y-axis, so it is possible to consider only oneside of the profile P. In the following description, the left half ofthe profile P is considered. Points on the profile are selected atregular intervals Δy along y-axis, where Δy is much smaller than theheight of the profile. This selection allows the continuous curve to beapproximated with discrete points denote p1, p2, . . . pn. In otherwords, the points p1, p2, . . . pn correspond to the coordinates (x1,0), (x2, Δy), . . . (xn, (n−1) Δy), respectively. These points p1, p2, .. . , pn from the base set B and the sets in C correspond to therectangles that can be generated by these points.

As shown in the exemplary rectangularization of FIG. 5, each rectanglehas its bottom left corner at point pi from B, and its top left cornerhas the same x-coordinate as its bottom left coordinate. Additionally,the y-coordinate of its top left corner is some value jΔy, where j≦i.Thus, there are (n*(n−1))/2 different rectangles that can be formed byselecting two heights iΔy and jΔy along the profile—these rectangleshave all possible heights from Δy to nΔy, and all possible positions inthe profile P as long as the top and bottom of the rectangle lies within(or at the top or bottom edges of) the profile P. These rectangles aredenoted by R1, R2, . . . , Rm, where m=(n*(n−1))/2. With reference toFIG. 6, the left-hand edge of rectangle Ri, which extends verticallyfrom rΔy to sΔy, where r and s are integers such that 0≦r<s≦n,approximates a subregion of P denoted by Si, and the set Ci includes allthe points of the profile P that lie within Si, i.e., all points on theprofile P with y coordinates between rΔy and sΔy.

Thus, a set system C that has its sets C1, C2 . . . . Cm is established.Weights are then assigned to the sets Ci. Since the objective of aset-cover problem is to minimize the total cost of the cover, theweights Wi are assigned to reflect that goal, i.e., to approximate theprofile shape by quantifying the quality of approximation. Therefore, asshown in FIG. 6, the weight Wi assigned to rectangle Ri is thedifference in area between the area of rectangle Ri and the area betweensection Si of the profile P and the y axis 605. As shown in FIG. y,where section Si lies outside of rectangle Ri, the weight Wi isconsidered to be a positive number. The larger the weight Wi/|Ci|, where|Ci| denotes the cardinality of set Ci, the worse the rectangle Ri is asan approximation of the profile P.

Thus far the mapping between a set-cover problem and therectangularization of the profile has been presented. The next step isto solve the set-cover problem. It has been shown that solving aset-cover problem is computationally difficult since the running time ofthe best known exact-solution algorithm is an exponential function ofthe input size. However, there are a number of efficient heuristics thatcan generate near-optimal solutions.

For example, a heuristic called a “greedy” heuristic can be used. Atevery step, this heuristic selects the set Ci whose value of Wi/|Ci| isthe least. It then adds Ci to the solution set Z and deletes all theelements in Ci for the base set B, and deletes any other sets Cj thatshare any elements with Ci. Additionally, any empty set in C is removedfrom it. Thus, at every step, the number of elements in the base set Bdecreases. This process is repeated until the base set B is empty. Atthis point, the solution set Z consists of sets that cover all theprofile points pi. The sets in the solution Z can be transformed backinto the rectangles which approximate the profile P. It should be notedthat the value of |Ci| at a given stage is the number of elements thatit contains in that stage—not the number of elements that it originallystarted with. Since the selection of sets Ci depends on the valueWi/|Ci|, the rectangles that are obtain can have different sizes. Adetailed description on the basic algorithm of this heuristic can befound in an article entitled “Approximation algorithms for clustering tominimize the sum of diameters.” by Srinivas Doddi, Madhav Marathe, S. S.Ravi, David Taylor, and Peter Widmayer, Scandinavian workshop onalgorithm theory (SWAT) 2000, Norway, the entire content of which isincorporated herein by reference.

Although the above method returns a set of rectangles that approximate agiven profile, the number of rectangles might be very large. In theabove mentioned article, Doddi, et. al found that by uniformlyincreasing the weights of each set by Δw and remaining the above method,the number of rectangles will be reduced. By repeating this process forincreasing values of Δw, it is possible to achieve a target number ofrectangles.

Although rectangles have been described as being used to representprofile shapes, it should be noted that any other geometric shape,including trapezoids, can be used. A process for automaticallyapproximating a profile with trapezoids may, for instance, be applied tothe step of adjusting the range to vary the parameters in generating aset of simulated-diffraction signals.

7. Determining the Number of Harmonic Orders to Use in Generating theSet of Simulated-Diffraction Signals

As described above, in the present exemplary embodiment,simulated-diffraction signals can be generated using a rigorous coupledwave analysis (RCWA). For a more detailed description of RCWA, see T. K.Gaylord, M. G. Moharam, “Analysis and Applications of OpticalDiffraction by Gratings”, Proceedings of the IEEE, vol. 73, no. 5, May1985, the entire content of which is incorporated herein by reference.

Prior to performing an RCWA calculation, the number of harmonic ordersto use is selected. In the present exemplary embodiment, an OrderConvergence Test is performed to determine the number of harmonic ordersto use in the RCWA calculation. More particularly, simulated-diffractionsignals are generated using RCWA calculations with the number ofharmonic orders incremented from 1 to 40 (or higher if desired). Whenthe change in the simulated-diffraction signal for a pair of consecutiveorder values is less at every wavelength than the minimum absolutechange in the signal that can be detected by the optical instrumentationdetector (e.g., detector 170 in FIG. 1), the lesser of the pair ofconsecutive orders is taken to be the optimum number of harmonic orders.

When multiple profile shapes are determined in characterizing periodicgrating 145 (FIG. 1), an Order, Convergence Test can be performed foreach of these profile shapes. In this manner, the maximum number ofharmonic orders obtained from performing the Order Convergence Test isthen used in generating library 185 (FIG. 1).

8. Determining a Resolution to Use in Generating the Set ofSimulated-Diffraction Signals

As described earlier, the value of hypothetical parameters are variedwithin a range to generate a set of hypothetical profiles.Simulated-diffraction signals are then generated for the set ofhypothetical profiles. Each simulated-diffraction signal is paired witha hypothetical profile, then the pairings are stored in library 185(FIG. 1). The increment at which the hypothetical parameters are varieddetermines the library resolution of library 185 (FIG. 1). As such, thesmaller the increment, the finer the resolution, and the larger the sizeof the library.

Thus, the resolution of hypothetical parameters used in generatinglibrary 185 (FIG. 1) is determined to provide a compromise between (1)the minimization of the size of the library by using large libraryresolutions, and (2) providing accurate matches between signals andprofiles by using small library resolutions. More particularly, in thepresent exemplary embodiment, an abbreviated library is generated usinga portion of the range used to generate the full library. Using theabbreviated library, the lowest resolution is determined for thehypothetical parameters that do not have specified resolutions thatstill provide accurate matches for the critical parameters.

By way of example, assume that three hypothetical parameters (top CD,middle CD, and bottom CD) are used to characterize a profile. Assumethat the range for the top CD, middle CD, and bottom CD are 60 to 65nanometers, 200 to 210 nanometers, and 120 to 130 nanometers,respectively. Also assume that the critical parameter is the bottom CDand the specified resolution for the bottom CD is 0.1 nanometer, and noparticular resolution is specified for the top) and middle CDs.

In the present exemplary embodiment, an abbreviated library is generatedusing a portion of the range specified for the hypothetical parameters.In this example, an abbreviated library of simulated-diffraction signalsis generating for top CD between 60 and 61, middle CD between 200 and201, and bottom CD between 120 and 121.

Initially, the abbreviated library is generated at the highest specifiedresolution. In this example, simulated-diffraction signals are generatedfor the top CD, middle CD, and bottom CD as they are incremented by 0.1nanometers between their respective ranges. For example,simulated-diffraction signals are generated for a top CD of 60, 60.1,60.2, . . . , 60.9, and 61. Simulated diffraction signals are generatedfor middle CD of 200, 200.1, 200.2, . . . , 200.9, and 201.Simulated-diffraction signals are generated for bottom CD of 120, 120.1,120.2, . . . . 120.9, and 121.

The resolution of the non-critical parameters is then incrementallyreduced in the abbreviated library until an attempted match for thecritical parameter fails. In this example, the simulated-diffractionsignal corresponding to the set of hypothetical parameters with top CDof 60.1, middle CD of 200, and bottom CD of 120 is removed from theabbreviated library. An attempt is then made to match the removedsimulated-diffraction signal with remaining simulated-diffractionsignals in the abbreviated library. If a match is made with asimulated-diffraction signal having the same critical parameter as theremoved simulated-diffraction signal (i.e., a bottom CD of 120), thenthe resolution for the top CD can be further reduced. In this manner,each of the non-critical parameters are tested to determine the minimumresolution that can be used. This study is performed for all thenon-critical parameters simultaneously in order to take into account theparameter interaction effects.

In the following description, a more thorough description is provided ofa process for determining the resolution Δp_(i) of hypotheticalparameters p_(i) used in generating library 185 (FIG. 1) is determinedto provide a compromise between (1) the minimization of the size of thelibrary by using large library resolutions Δp_(i), and (2) providingaccurate matches between signals and profiles by using small libraryresolutions Δp_(i).

The parameters p_(i) which are used to characterize various profiles Pwere described in detail above. In the following description, thegeneral case of m parameters p₁, p₂, . . . ,p_(m) will be presented, andthe special case of m=2 will be depicted in FIG. 9 and presented in textenclosed in curly brackets “{ }”. {For concreteness, consider the firstparameter p₁ to be the width w1 of a rectangular profile, and the secondparameter p₂ to be the height h1 of a rectangular profile.} Therefore,any profile P may be represented by a point in an ne-dimensional space.{Therefore, as shown in FIG. 9, any profile P may be represented by apoint in a two-dimensional space.} The range of profiles P to be used inlibrary 185 (FIG. 1) may be specified by setting minimum and maximumvalues of each parameter p_(i) ^((min)) and p_(i) ^((max)).

Typically, the particular resolution of interest in semiconductorfabrication, i.e., the target resolution R, is the resolution of thecritical dimension. In general, the resolution of the critical dimensionis some function of the resolution Δp_(i) of multiple parameters p_(i).{In the two-dimensional case, the resolution of the critical dimensionhappens to be the resolution Δp₁ of the first parameter p₁=w₁. But tomake the two-dimensional discussion correspond to the general case, thecritical dimension will be assumed to be a function of the resolutionΔp_(i) of multiple parameters p_(i).}

Typically only a single target resolution R is considered. However, inthe present embodiment, multiple target resolutions R_(i) can beconsidered, and the accuracy of the mappings between profiles andsignals allows the resolution Δp_(i) of multiple profile shapeparameters p_(i) to be determined.

A grating of a particular profile P produces a complex-valueddiffraction signal S(P, λ), is plotted as a function of wavelength λ.The magnitude of the signal S(P, λ) is the intensity, and the phase ofthe signal S(P, λ) is equal to the tangent of the ratio of two,perpendicular planar polarizations of the electric field vector. Adiffraction signal may, of course, be digitized, and the sequence ofdigital values may be formed into a vector, albeit a vector having alarge number of entries if the signal is to be accurately represented.Therefore, each signal S(P, λ) corresponds to a point in ahigh-dimensional signal space, and points in the high-dimensional spacewhich are near each other correspond to diffraction signals which aresimilar. For ease of depiction in the present discussion, in FIG. 10 asignal space with a dimensionality of two, s₁ and s₂, is shown. Thetwo-dimensional depiction of FIG. 10 may be considered to be aprojection of the high-dimensionality signal space onto two dimensions,or a two-dimensional slice of the signal space.

In the present embodiment, the determination of the library resolutionsΔp_(i) of the parameters p_(i) begins by choosing a nominal profilep^((n)) and generating its corresponding signal S(P^((n))). Then a setof profiles P near the nominal profile P^((n)) is generated. This may bedone by choosing a regularly-spaced array of points in the profile spacearound the nominal n, an irregularly-spaced array of points in theprofile space around the nominal n, or a random scattering of points inthe profile space around the nominal n. For ease of discussion anddepiction, a regularly-spaced array of points around the nominal n willbe considered {and depicted in FIG. 9}, so parameter increment valuesδp_(i) are chosen for each parameter p_(i). Therefore, profiles locatedatn+Σ _(i)α_(i) δp _(i),and the corresponding diffraction signalsS(n+Σ _(i)α_(i) δp _(i))are generated, where α₁ takes integer values ( . . . , −2, −1, 0, 1, 2,3, . . . ) and the sum runs from i=1 to i=m, and n is the vectorcorresponding to the nominal profile p^((n)). {Therefore, as shown inFIG. 9, profiles located atn+α ₁ δp ₁+α₂ δp ₂,and the corresponding diffraction signals,S(n+α ₁ δp ₁+α₂ δp ₂),are generated, where α₁ and α₂ take integer values ( . . . , −2, −1, 0,1, 2, 3, . . . ).} (For ease of presentation, a profile p and itscorresponding vector in the profile space will be used synonymously.)The parameter increment values δp_(i) are chosen to be small relative tothe expected values of the library resolutions Δp_(i), i.e.,δp _(i) >>Δp _(i).{In the example shown in FIG. 9, the parameter increment values δp₁ andδp₂ are chosen to be one-eighth and one-sixth the size of the ranges (p₁^((max))−p₁ ^((min))) and (p₂ ^((max))−p₂ ^((min)), respectively, ofparameter values used in determining the resolutions of the parameters.}In practice, the parameter increment values δp_(i) are chosen to beorders of magnitude smaller than the sizes of the ranges (p₁ ^((max))−p₁^((min))) and resolutions Δp_(i) of parameter values. While the profilesP may be selected to correspond to points on a grid, generally thecorresponding diffraction signals S, which are depicted as dots in FIG.10, are not located at regularly spaced intervals.

The next step in determining the resolutions Δp_(i) of the parametersp_(i), is to order the signals S(n+α₁δp₁+α₂δp₂) by increasing distancefrom the signal S(n) of the nominal profile p^((n)), which willhereafter be referred to as the nominal signal S(n) or S^((n)). In thepresent embodiment, the distance between a first signal S⁽¹⁾ and asecond signal S⁽²⁾ is measured using a sum-squared-difference-log errormeasure φ, i.e.,φ(S ⁽¹⁾ , S ⁽²⁾=Σ_(λ)[log S ⁽¹⁾(λ)−log S ⁽²⁾(λ)]²,where the sum is taken over uniformly-spaced wavelengths λ. As shown inFIG. 10, this is graphically represented by drawing a series of closelyspaced hyperspheres, which are represented in FIG. 10 as circles 1002,1004, 1006, and 1008, centered around the nominal signal S(n), andordering the signals S(n+Σ_(i)α_(i)δp_(i)) {S(n+α₁δp₁+α₂δp₂)} accordingto the largest hypersphere 1002, 1004, 1006, and 1008 which encloseseach signal S(n+Σ_(i)α_(i)δp_(i)) {S(n+α₁δp₁+α₂δp₂)}. The smallesthypersphere 1002 corresponds to the resolution ε of the instrumentation,i.e., all signals S within the smallest hyperspace 1002 satisfyS ^((n))(λ)−S(λ)≦ε,at all wavelengths λ. In the exemplary case of FIG. 10, four signals areshown to be within circle 1002.

According to the next step of the present invention, the signalsS(n+Σ_(i)α_(i)δp_(i)) {S(n+α₁δp₁+α₂δp₂)} are tested in order ofincreasing distance φ from the nominal signal S^((n)) to determine whichis the signal S(n+Σ_(i)α_(i)δp_(i)) {S(n+α₁δp₁+α₂δp₂)} closest to thenominal signal S^((n)) which has a profile (n+Σ_(i)α_(i)δp_(i)){n+α₁δp₁+α₂δp₂} which differs from the nominal profile P^((n)) by thetarget resolution R. In the case of multiple target resolutions R, thesignals S(n+Σ_(i)α_(i)δp_(i)) {S(n+α₁δp₁+α₂δp₂)} are tested in order ofincreasing distance φ from the nominal signal S^((n)) to determine whichis the signal S(n+Σ_(i)α_(i)δp_(i)) {S(n+α₁δp₁+α₂δp₂)} closest to thenominal signal S^((n)) which has a profile (n+Σ_(i)α_(i)δp_(i)){n+α₁δp₁+α₂δp₂} which differs from the nominal profile P^((n)) by one ofthe target resolutions R_(i). That particular signal is termed theborder signal S^((B)), and the smallest hypersphere 1002, 1004, 1006,and 1008 which encloses the border signal S^((B)) is termed the borderhyperspace B. For those signals S which fall outside the borderhyperspace B, the corresponding profiles P are discarded fromconsideration in the process of determining the library resolutionsΔp_(i).

Then, for each signal S which lies within the border hyperspace B, adisplacement vector V for its relation to the nominal profile vector nis determined. In particular, the displacement vector V between aprofile P^((α)) described by the vector (p₁ ^(α), p₂ ^(α), . . . p_(m)^(α)) and the nominal vector n=(p₁ ^(n), p₂ ^(n), . . . p_(m) ^(n)) isgiven byV=(p ₁ ^(α) −p ₁ ^(n) −p ₂ ^(α) −p ₂ ^(n) , . . . p _(m) ^(α) −p _(m)^(n)),{or in the two-dimensional case depicted in FIG. 9,V=(p ₁ ^(α) −p ₁ ^(n) , p ₂ ^(α) −p ₂ ^(n)).The exemplary displacement vector V shown in FIG. 9 is V=(1,2).} The setof equivalent displacement vectors V′ is defined asV′=(±|p ₁ ^(α) −p ₁ ^(n) |, ±|p ₂ ^(α) −p ₂ ^(n) |, . . . , ±|p _(m)^(α) −p _(m) ^(n)|).{or in the two-dimensional case depicted in FIG. 9,V′=(±|p ₁ ^(α) −p ₁ ^(n) |, ±p ₂ ^(α) −p ₂ ^(n) |),}i.e., the set of equivalent displacement vectors V′, which includes theoriginal displacement vector V, defines the 2^(m) {four} corners of anm-dimensional hyperrectangle {a two-dimensional rectangle 920 depictedin FIG. 9}.

Then, for each signal S(V) which lies within the border hypersphere B,it is determined whether all the equivalent displacement vectors V′correspond to signals S(V′) which also lie within the hypersphere B. Ifone or more signals S(V′) do not lie within the hypersphere B, theprofiles corresponding to the entire set of the equivalent displacementvectors V′ are discarded from consideration in the process ofdetermining the library resolutions Δp_(i). In other words, what remainsunder consideration in determining the library resolutions Δp_(i) arethose m-dimensional hyperrectangles {two-dimensional rectangles} inprofile space for which all the corresponding signals S lie inside theborder hypersphere B. It is these m-dimensional hyperrectangles{two-dimensional rectangles} which are under consideration as thelibrary resolutions Δpi.

For each of the m-dimensional hyperrectangles {two-dimensionalrectangles} in profile space for which all the corresponding signals Slie inside the border hypersphere B, the number N of m-dimensionalhyperrectangles {two-dimensional rectangles} required to fill theprofile space is simulated. For a p₁*×p₂*× . . . ×p_(m)* hyperrectangle,the count number N is the number of such p₁*×p₂*× . . . ×p_(m)*hyperrectangles which fit into a hyperrectangular space defined by thebounds p_(i) ^((min))<p_(i)<p_(i) ^((max)). The count number N is givenbyN=max[p ₁ ^((max)) −p ₁ ^((min)) /p ₁*, (p ₂ ^((max)) −p ₂ ^((min)) /p₂*, . . . ],where the square brackets in the above equation indicate that eachfractional value within is rounded up to the nearest integer. {Forinstance, for the 2δp₁×4δp₂ rectangle 620 defined by the equivalentvectors V′ shown in FIG. 9, since the rectangular profile space has awidth of (p₁ ^((max))−p₁ ^((min)))=9δp₁ and a height of (p₂ ^((max))−p₂^((min))=6δp₂, the count number N is five.}

Finally, the resolutions Δp_(i) which are used for the library are equalto the dimensions of the m-dimensional hyperrectangular defined by theset of equivalent vectors V′, which (i) has the smallest count N, andfor which (ii) all the corresponding signals S(V′) lie inside the borderhypersphere B.

9. Generating the Set of Simulated-Diffraction Signals Based on theAdjusted Range, Parameterization, and/or Resolution

In the present exemplary embodiment, library 185 (FIG. 1) is generated,wherein both the profile shape and the film geometry (thickness andwidth) parameters are varied using the adjusted parameterization, rangesand resolutions determined above. As such, the number of profilesgenerated in library 185 (FIG. 1) is a function of the profile shapeparameterization and the ranges and resolutions of the parameters.Additionally, the library entries are a function of grating pitch,optical properties of films in the underlying and patterned layers,profile parameter ranges, profile parameter resolutions and profileshapes. It should be noted that library 185 (FIG. 1) can be generated byusing only the adjusted ranges or only the adjusted resolution.

10. Comparing a Set of Measured-Diffraction Signals with theSimulated-Diffraction Signals in the Library

In the present exemplary embodiment, after generating library 185 (FIG.1), a set of measured-diffraction signals are compared with thesimulated-diffraction signals in library 185 (FIG. 1), as a qualitycontrol. If the error between the best match found in library 185(FIG. 1) and the measured-diffraction signal is better than a thresholdgoodness-of-fit limit, then the library generation process is consideredsuccessful. Alternatively, and more preferably, a quality control can beinsured by comparing the width and height values obtained using anothermeasurement technique, such as an X-SEM, CD-SEM, and the like.

Although exemplary embodiments have been described, variousmodifications can be made without departing from the spirit and/or scopeof the present invention. Therefore, the present invention should not beconstrued as being limited to the specific forms shown in the drawingsand described above.

1. A method of generating a library of simulated-diffraction signals(simulated signals) of a periodic grating, said method comprising:obtaining a measured-diffraction signal (measured signal) of theperiodic grating; associating hypothetical parameters with ahypothetical profile; varying the hypothetical parameters within a rangeto generate a set of hypothetical profiles; adjusting the range to varythe hypothetical parameters based on the measured signal; and generatinga set of simulated signals from the set of hypothetical profiles.
 2. Themethod of claim 1 further comprising extracting optical properties ofthe periodic grating.
 3. The method of claim 2, wherein the periodicgrating is formed from a plurality of materials, each material having arefractive index, and wherein extracting optical properties includesextracting real and imaginary parts of the refractive index of eachmaterial.
 4. The method of claim 3, wherein the real and imaginary partsof the refractive index are extracted using a simulated annealing basedoptimizer.
 5. The method of claim 1 further comprising determining thenumber of harmonic orders to use in generating the set of simulatedsignals.
 6. The method of claim 5, wherein determining the number ofharmonic orders includes running a convergence test.
 7. The method ofclaim 6 further comprising: generating simulated signals usingincreasing number of orders; determining the change in the simulatedsignals with the increase in the number of orders used; and selectingthe lower number of orders when the change in the simulated signal isless than the minimum change in the measured signal that can beobtained.
 8. The method of claim 1 further comprising: dividing thehypothetical profile into a plurality of hypothetical layers; anddetermining the number of hypothetical layers to use in generating theset of simulated signal for the hypothetical profile, wherein eachhypothetical profile in the set of hypothetical profiles can be dividedinto different numbers of hypothetical layers.
 9. The method of claim 8,wherein determining the number of hypothetical layers includes: mappingthe determination of the number of hypothetical layers as a set-coverproblem; and solving the set-cover problem.
 10. The method of claim 1,wherein the periodic grating includes a first layer formed on asubstrate and a second layer formed on the first layer, and wherein theobtaining measured diffraction signal of the periodic grating includes:measuring a first-diffraction signal after forming the first layer onthe substrate prior to forming the second layer on the first layer; andmeasuring a second-diffraction signal after forming the second layer onthe first layer.
 11. The method of claim 1, wherein a plurality ofmeasured signals is obtained from a plurality of sites on asemiconductor wafer.
 12. The method of claim 11, wherein a plurality ofmeasured signals is obtained from a plurality of semiconductor wafers.13. The method of claim 11, wherein adjusting the range to vary thehypothetical parameters includes: comparing simulated signals and themeasured signals using an error metric; and shifting the range to varythe hypothetical parameters when the simulated signals and the measuredsignals match and when the hypothetical parameters of the simulatedsignals lie near an upper or lower limit of the range.
 14. The method ofclaim 13, wherein the error metric is a sum-squared error.
 15. Themethod of claim 13, wherein the error metric is asum-squared-difference-log error.
 16. The method of claim 1 furthercomprising: determining a resolution for the set of simulated signals;and varying the hypothetical parameters used in generating the simulatedsignals at an increment corresponding to the determined resolution. 17.The method of claim 16, wherein the resolution for the parameters isdetermined based on a desired critical dimension of the periodicgrating.
 18. The method of claim 17, wherein determining the resolutionfor the parameters further comprises: generating a sub-set of simulatedsignals including: a first-simulated signal generated using a first setof hypothetical parameters, wherein the first set of hypotheticalparameters includes: a first-hypothetical parameter associated with thedesired critical dimension, and a second-hypothetical parameter notassociated with the desired critical dimension, and a second-simulatedsignal generated using a second set of hypothetical parameters, whereinthe second set of hypothetical parameters includes: a first-hypotheticalparameter matching the first-hypothetical parameter of thefirst-simulated signal, and a second-hypothetical parameter notassociated with the desired critical dimension and not matching thesecond-hypothetical parameter of the first-simulated signal; removingthe second-simulated signal from the sub-set of simulated signals;comparing the second-simulated signal to the remaining simulated signalsin the sub-set of simulated signals; and reducing the resolution to usefor the second-hypothetical parameter in generating the set of simulatedsignals if the comparison matches the second-simulated signal to thefirst-simulated signal.
 19. The method of claim 1, wherein associatinghypothetical parameters with a hypothetical profile further comprises:determining the number of hypothetical parameters to associate with thehypothetical profile based on the measured signal.
 20. The method ofclaim 19, wherein determining the number of hypothetical parametersfurther comprises: generating a set of simulated signals using thedetermined number of hypothetical parameters; comparing the measuredsignal to the set of simulated signals; and increasing the number ofhypothetical parameters if the measured signal fails to match any of thesimulated signals in the set of simulated signals. 21-65. (canceled)